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Document Neo-Hookean model #608

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#605
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Document Neo-Hookean model#608
#605
@mvanzulli

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@mvanzulli
mvanzulli
opened on Feb 7, 2023

Considering this strain energy function is:

$$\Psi(\mathbf{C}) = \frac{\mu}{2}(I_1 -3) + 1/D (J -1)^2$$

then Cosserat is:

$$\mathbf{S} = 2 \frac{\partial \Psi }{\partial \mathbf{C}} = \mu (I_{3,3}) + \frac{2}{D}(J-1) J C^{-T}$$

but in ONSAS we have:

S       = shear * ( eye(3) - invC ) + bulk * ( J * (J-1)* invC) ;

Useful identities:

$$J = \sqrt(det(\mathbf{C}))$$ $$\frac{\partial J}{\partial \mathbf{C}} = \frac{J \mathbf{C}^{-T}}{2}$$ $$\frac{\partial I_1}{\partial \mathbf{C}} = I_{3,3}$$

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